Summary: We consider the propagation of a flame front in a solid periodic medium. The model is governed by a free boundary system in which the front’s velocity depends on the temperature via an Arrhenius kinetic. We show the existence of travelling wave solutions and consider their homogenization as the period tends to zero. The main difficulty lies in the degeneracy of the Arrhenius function which requires an a priori lower bound of the propagation’s speed. We next analyze the curvature effects on the homogenization and obtain a continuum of limiting waves parametrized by the ratio “curvature coefficient/period.” Remarkable features are the monotonicity of the speed with respect to the “curvature regime,” together with the explicit computations of the minimal and maximal speeds. We finally identify the asymptotic expansion of the heterogeneous front’s profile with respect to the period.